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Transport and localization of waves in ladder-shaped lattices with locally $\mathcal{PT}$-symmetric potentials

We study numerically the transport and localization properties of waves in ordered and disordered ladder-shaped lattices with local $\mathcal{PT}$ symmetry. Using a transfer matrix method, we calculate the transmittance and the reflectance for the individual channels and the Lyapunov exponent for the whole system. In the absence of disorder, we find that when the gain/loss parameter $ρ$ is smaller than the interchain coupling parameter $t_{v}$, the transmittance and the reflectance are periodic functions of the system size, whereas when $ρ$ is larger than $t_{v}$, the transmittance is found to be an exponentially-decaying function while the reflectance attains a saturation value in the thermodynamic limit. For a fixed system size, there appear perfect transmission resonances in each individual channel at several values of the gain/loss strength smaller than $t_{v}$. A singular behavior of the transmittance is also found to appear at various values of $ρ$ for a given system size. When disorder is inserted into the on-site potentials, these behaviors are changed substantially due to the interplay between disorder and the gain/loss effect. When $ρ$ is smaller than $t_{v}$, we find that the presence of locally $\mathcal{PT}$-symmetric potentials suppresses Anderson localization, as compared to the localization in the corresponding Hermitian system. When $ρ$ is larger than $t_{v}$, we find that localization becomes more pronounced at higher gain/loss strengths. We also find that the phenomenon of anomalous localization occurs in disordered locally $\mathcal{PT}$-symmetric systems precisely at the spectral positions $E=0$ and $E=\pm\sqrt{t_{v}^2-ρ^2}$. The anomaly at the band center manifests as a sharp peak contrary to the conventional cases, whereas the anomalies at $E=\pm\sqrt{t_{v}^2-ρ^2}$ manifest as sharp dips.